# Are these three graphs isomorphic?

I believe that these three graphs are non-isomorphic, because the first graph contains a 6 cycle, which is not present in the other two graphs. The 2nd and 3rd graphs are not isomorphic because they have different numbers of 5 cycles.

• Is it possible to align the 2nd image in the center? – Sabhrant Jan 27 at 16:20
• How many $5$-cycles have you counted in the second and third graphs? – Blue Jan 27 at 16:31
• Keep counting in the second graph. – Blue Jan 27 at 16:34
• Right; same number of cycles, so you have to try something else. (This is why the Graph Isomorphism Problem is hard.) Try matching the $5$-cycles from one graph to those of the other. For instance, the inner $5$-cycle of the second graph would be a nice match for one of the "inner" $5$-cycles of the third; go from there. (It would be convenient if you could get the third graph's other "inner" $5$-cycle out of the way ... moving its vertices to the outside, for instance ...) – Blue Jan 27 at 16:41
• The 2nd and 3rd graph are both a ''2D version'' of the dodecahedron actually. – Colorblind97 Jan 27 at 16:51